Related papers: Spectral mapping theorems of differentiable C0 sem…

We establish spectral inclusion and mapping theorems for scalar type spectral operators, generalizing their counterparts for normal operators. Thereby, we extend a precise weak spectral mapping theorem, known to hold for $C_0$-semigroups of…

We study linear operators $T$ on Banach spaces for which there exists a $C_0$-semigroup $(T(t))_{t\geq 0}$ such that $T=T(1)$. We present a necessary condition in terms of the spectral value 0 and give classes of examples where this can or…

Let $A$ and $B$ be unital semisimple commutative Banach algebras and $T$ a map from the invertible group $A^{-1}$ onto $B^{-1}$. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on $T$: (1) $\sigma…

Let $ (T(t))_{t\geq0} $ be a positive $C_{0}$-semigroup with generator $A$ on a C$^*$-algebra or on the predual of a W$^*$-algebra. Then the growth bound $\omega_{0}$ equals $s(A)$. If the spectrum of $A$ is not empty, then $s(A)$, the…

The well-known Batty's theorem states that if a $C_0$-semigroup $T(t)$ is bounded and the spectrum of the generator $A$ is contained in the open left-half plane of $\mathbb{C}$, then $\|T(t)A^{-1}\|$ tends to $0$. This can be thought of as…

Given a positive, irreducible and bounded C_0-semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or…

Let $(T(t))_{t\geq0}$ be a $C_0$ semigroups and $A$ be its infinitesimal generator. In this work, we prove that the spectral inclusion for $(T(t))_{t\geq0}$ remains true for the Drazin invertible and Quasi-Fredholm spectra. Also, we will…

A bounded operator $T$ in a Banach space $X$ is said to satisfy the essential descent spectrum equality, if the descent spectrum of $T$ as an operator on $X$ coincides with the essential descent spectrum of $T$. In this note, we give some…

Let $A$ be a Banach algebra. By $\sigma(x)$ and $r(x)$ we denote the spectrum and the spectral radius of $x\in A$, respectively. We consider the relationship between elements $a,b\in A$ that satisfy one of the following two conditions: (1)…

Let ${\bf R}$ denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear…

We introduce the numerical spectrum $\sigma_n(A)\subset \mathbb{C}$ of an (unbounded) linear operator $A$ on a Banach space $X$ and study its properties. Our definition is closely related to the numerical range $W(A)$ of $A$ and always…

We present a spectral mapping theorem for semigroups on any Banach space $E$. From this, we obtain a characterization of exponential dichotomy for nonautonomous differential equations for $E$-valued functions. This characterization is given…

In this paper several joint spectra for a finite commuting family of closed operators in Banach space are considered, some new relations between these spectra established (earlier only the inclusion of the Taylor spectrum in the commutant…

Let $\{T(t)\}_{t\geq 0}$ be a $C_0$-semigroup of bounded linear operators on the Banach space ${X}$ into itself and let $A$ be their infinitesimal generator. In this paper, we show that if $T(t)$ is uniformly ergodic, then $A$ does not have…

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the…

In this paper, we give conditions for which the $C_0$ semigroups satisfies spectral equality for semiregular, essentially semiregular and semi-Fredholm spectrum. Also, we establish the spectral inclusion for B-Fredholm spectrum of a $C_0$…

A basic result in semigroup theory states that every $C_0$-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the…

Gramsch and Lay [10] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$, $$\widetilde{\sigma}_i(f(T)) = f(\widetilde{\sigma}_i(T)), $$ for several extended essential spectra…

Let $X$ be a Banach space, and $T:[0,\infty)\rightarrow {\mathcal{L}}(X,X),$ the bounded linear operators on $X.$ A family $\{T(t)\}_{t\ge 0}\subseteq {% \mathcal{L}}(X,X)$ is called a one-parameter semigroup if $T(s+t)=T(s)T(t),$ and…

Let $A$ be a complex semisimple Banach algebra with identity, and denote by $\sigma'(x)$ and $\rho (x)$ the nonzero spectrum and spectral radius of an element $x \in A$, respectively. We explore the relationship between elements $a, b \in…